Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13

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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13

Choose the correct or most suitable answer from given four alternatives.
Question 1.
If \(\int f(x) d x\) = g(x) + c, then \(\int f(x) g^{\prime}(x) d x\)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 1
Solution:
(a)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 2

Question 2.
If Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 3, then the value of k is ……………
(a) log 3
(b) -log 3
(c) \(-\frac{1}{\log ^{3}}\)
(d) \(\frac{1}{\log 3}\)
Solution:
(c)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 4

Question 3.
If \(\int f^{\prime}(x) e^{x^{3}} d x\) = (x – 1)ex2, then f(x) is …………………
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 5
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 6

Question 4.
The gradient (slope) of a curve at any point (x, y) is \(\frac{x^{2}-4}{x^{2}}\). If the curve passes through the point(2, 7), then the equation of the curve is ………….
(a) y = x + \(\frac{4}{x}\) + 3
(b) y = x + \(\frac{4}{x}\) + 4
(c) y = x2 + 3x + 4
(d) y = x2 – 3x + 6
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 7

Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13

Question 5.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 8
(a) cot (xex) + c
(b) sec (xex) + c
(c) tan (xex) + c
(d) cos (xex) + c
Solution:
(c)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 9

Question 6.
\(\int \frac{\sqrt{\tan x}}{\sin 2 x} d x\) is ……………..
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 10
Solution:
(a)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 11

Question 7.
\(\int \sin ^{3} x d x\) is …………….
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 12
Solution:
(c)
Hint: sin3x = \(\frac{1}{4}\) (3 sin x – sin 3x)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 13

Question 8.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 14
Solution:
(b)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 15

Question 9.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 16
(a) tan-1 (sin x) + c
(b) 2 sin-1 (tan x) + c
(c) tan-1 (cos x) + c
(d) sin-1 (tan x) + c
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 17
= sin-1 (t) + c
= sin-1 (tan x) + c

Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13

Question 10.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 18
(a) x2 + c
(b) 2x2 + c
(c) \(\frac{x^{2}}{2}\) + c
(d) \(-\frac{x^{2}}{2}\) + c
Solution:
(c)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 19

Question 11.
\(\int 2^{3 x+5} d x\) is ……………
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 20
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 21

Question 12.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 22
Solution:
(b)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 23

Question 13.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 24
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 25

Question 14.
\(\int \frac{x^{2}+\cos ^{2} x}{x^{2}+1}\) cosec2xdx is …………….
(a) cot x + sin-1 x + c
(b) -cot x + tan-1 x + c
(c) -tan x + cot-1 x + c
(d) -cot x – tan-1 x + c
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 26

Question 15.
\(\int x^{2} \cos x d x\) is ……………
(a) x2 sin x + 2x cos x – 2 sin x + c
(b) x2 sin x – 2x cos x – 2 sin x + c
(c) -x2 sin x + 2x cos x + 2 sin x + c
(d) -x2 sin x – 2x cos x + 2 sin x + c
Solution:
(a)
Hint: \(\int x^{2} \cos x d x\)
By Bernoullis formula dv = cosxdx
u = x2 v = sinx
u’ = 2x v1 = -cos x
u” = 2 v2 = -sinx
= uv – u’v1 + u”v2
= x2sin x + 2x cos x – 2 sin x + c

Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13

Question 16.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 27
Solution:
(b)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 28

Question 17.
\(\int \frac{d x}{e^{x}-1}\) is …………….
(a) log |ex| – log |ex – 1| + c
(b) log |ex| + log |ex – 1| + c
(c) log |ex – 1| – log |ex| + c
(d) log |ex + 1| – log |ex| + c
Solution:
(c)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 29

Question 18.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 30
Solution:
(b)
We know that
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 31

Question 19.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 32
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 33

Question 20.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 34
Solution:
(a)
We know that
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 35

Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13

Question 21.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 36
Solution:
(c)
Hint:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 37
By Bernoullis formula,
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 38

Question 22.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 39
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 40

Question 23.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 41
Solution:
(c)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 42
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 43

Question 24.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 44
Solution:
(a)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 45

Question 25.
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 46
Solution:
(d)
Hint: Let I = \(\int e^{\sqrt{x}} d x\)
t = \(\sqrt{x}\)
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13 47